© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11n325 L11n327

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n326 Visit L11n326's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18

Gauss Code:

{{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}}

Jones Polynomial:

- q-6 + 2q-5 - 2q-4 + 5q-3 - 4q-2 + 5q-1 - 3 + 3q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-22 - q-18 + q-16 + 2q-14 + 3q-12 + 5q-10 + 3q-8 + 5q-6 + 2q-4 + 3q-2 + 2 + q2 + q4 + q8

HOMFLY-PT Polynomial:

z-2 + 4 + 7z2 + 5z4 + z6 - 2a2z-2 - 9a2 - 18a2z2 - 17a2z4 - 7a2z6 - a2z8 + a4z-2 + 7a4 + 11a4z2 + 6a4z4 + a4z6 - 2a6 - a6z2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 4a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 2a-1z7 - z-2 + 3 - 3z2 + 7z4 - 8z6 + 2z8 + 2az-1 - 8az + 13az3 - 6az5 - 2az7 + az9 - 2a2z-2 + 11a2 - 28a2z2 + 37a2z4 - 21a2z6 + 4a2z8 + 2a3z-1 - 12a3z + 17a3z3 - 3a3z5 - 3a3z7 + a3z9 - a4z-2 + 11a4 - 27a4z2 + 28a4z4 - 12a4z6 + 2a4z8 - 7a5z + 11a5z3 - 5a5z5 + a5z7 + 3a6 - 5a6z2 + 2a6z4 - 2a7z + a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 1   j = 3               2 1   j = 1           1 2 1     j = -1         1 5 2       j = -3         3 4         j = -5       3 2           j = -7   1 1 3             j = -9   2 2               j = -11   1                 j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 326]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 326]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 21, 10, 20], X[19, 5, 20, 12], > X[11, 13, 12, 22], X[21, 11, 22, 10], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, > {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 2 5 4 5 2 3 -3 - q + -- - -- + -- - -- + - + 3 q - 2 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 -16 2 3 5 3 5 2 3 2 4 8 2 - q - q + q + --- + --- + --- + -- + -- + -- + -- + q + q + q 14 12 10 8 6 4 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 326]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 6 2 4 - 9 a + 7 a - 2 a + z - ---- + -- + 7 z - 18 a z + 11 a z - a z + 2 2 z z 4 2 4 4 4 6 2 6 4 6 2 8 > 5 z - 17 a z + 6 a z + z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 326]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 3 - a + 11 a + 11 a + 3 a - z - ---- - -- + --- + ---- - - - 8 a z - 2 2 z z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 > 12 a z - 7 a z - 2 a z - 3 z + ---- - 28 a z - 27 a z - 5 a z + 2 a 3 4 6 z 3 3 3 5 3 7 3 4 4 z 2 4 > ---- + 13 a z + 17 a z + 11 a z + a z + 7 z - ---- + 37 a z + a 2 a 5 6 4 4 6 4 8 z 5 3 5 5 5 6 z > 28 a z + 2 a z - ---- - 6 a z - 3 a z - 5 a z - 8 z + -- - a 2 a 7 2 6 4 6 2 z 7 3 7 5 7 8 2 8 > 21 a z - 12 a z + ---- - 2 a z - 3 a z + a z + 2 z + 4 a z + a 4 8 9 3 9 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 2 1 2 1 3 3 -- + - + q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 7 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t q t 2 3 1 2 t 2 3 2 3 3 5 3 7 4 > ---- + ---- + --- + --- + 2 q t + q t + 2 q t + q t + q t + q t 5 3 q t q q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n326

L11n325 L11n327